Discrimination and learning
Heidi L. Williams
MIT 14.662
Spring 2015
Williams (MIT 14.662)
Discrimination and learning
Spring 2015
1 / 50
Discrimination and learning
Previous models of statistical discrimination were static
In contrast, Altonji and Pierret (2001) use a dynamic model of
employer learning to test for statistical discrimination
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Theoretical model builds closely on Farber and Gibbons (1996)
Coate and Loury (1993) use a dynamic model of worker investments
to analyze affirmative action
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Discrimination and learning
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Roadmap for today
Preliminaries: Farber and Gibbons (1996)
Testing statistical discrimination: Altonji and Pierret (2001)
Affirmative action: Coate and Loury (1993)
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Discrimination and learning
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1
Preliminaries: Farber and Gibbons (1996)
2
Testing statistical discrimination: Altonji and Pierret (2001)
3
Affirmative action: Coate and Loury (1993)
4
Looking ahead
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Farber-Gibbons (1996)
At the time of labor market entry:
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Some characteristics (education) observed by employers, but likely
convey only partial information about productivity
Over time: worker gains experience, more information revealed
Key insight: the econometrician may observe variables measuring
productivity that are not observed by employers
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Example: AFQT scores
Can ask how employers learn about these over time
Farber-Gibbons: implications of employer learning for wages
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Influential framework
Tractable model
Empirically testable implications, generally supported by the data
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Model: Set-up
ηi : innate (time-invariant) ability
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Not observed by employers nor by econometrician
si : (time-invariant) schooling
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Observed by employers and by econometrician
Xi : (time-invariant) attributes other than schooling (race)
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Observed by employers and by econometrician
Zi : (time-invariant) attributes (school quality)
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Observed by employers; not observed by econometrician
Bi : (time-invariant) attributes (AFQT)
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Not observed by employers; observed by econometrician
Allow for arbitrary joint distribution F (ηi , si , Xi , Zi , Bi )
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Model: Set-up
yit : output of i in worker’s t th period in labor market
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{yit : t = 1, ..., T }: independent draws from conditional distribution
G (yit |ηi , si , Xi , Zi )
Note: Bi does not appear in this conditional distribution (assumes no
direct effect on output; can affect output via other variables, like ηi )
Assume:
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2
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Employers know F (ηi , si , Xi , Zi , Bi ) and G (yit |ηi , si , Xi , Zi )
Employers observe si , Xi , and Zi
Employers observe outputs {yi1 , ..., yit } through period t
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Strong “public learning” assumption
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Model: Set-up
Wage paid to a worker in period t is her expected output given all
available information available at t about the worker:
wit
= E (yit |si , Xi , Zi , yi1 , ..., yit−1 )
Spot-market model of wage determination
Rules out long-term contracts; strong assumption
Could be that long-term contracts are not useful, or that they are
useful but impossible to enforce
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Model: Predictions
Three predictions that can be tested in an earnings regression:
1
Effect of schooling on wages independent of experience
2
Time-invariant worker characteristics correlated with ability but
unobserved by employers increasingly correlated with wages as
experience increases
3
Wage residuals a martingale
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Prediction #1: The effect of schooling on wages
Consider a panel data set of one cohort of workers
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Data on si and Xi
Data on wage in each year (t = 1, 2, ..., T )
Can estimate the following earnings regression:
wit = αt + βt si + Xi γt + εit
Notes:
Zi by construction not included
Specified in levels, not logs
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Prediction #1: The effect of schooling on wages
Notation:
E ∗ (·): linear projection
E (·): conditional expectation
Estimated coefficients (αˆt , βˆt , γˆt ) are coefficients from linear projection
E ∗ (wit |si , Xi ) of wit on si and Xi :
E ∗ (wit |si , Xi ) = α̂t + βˆt si + Xi γˆt
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Prediction #1: The effect of schooling on wages
Recall:
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Version of law of iterated expectations: E ∗ (E (y |x, z)|x) = E ∗ (y |x)
[see notes]
wit = E (yit |si , Xi , Zi , yi1 , ..., yit−1 )
E ∗ (wit |si , Xi ) = E ∗ (E (yit |si , Xi , Zi , yi1 , ..., yit−1 ) |si , Xi )
= E ∗ (yit |si , Xi )
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Prediction #1: The effect of schooling on wages
Recall:
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si , Xi time-invariant
yit independent and identically distributed draws
⇒ E ∗ (yit |si , Xi ) is independent of t
⇒ effect of schooling on wages is independent of experience
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Prediction #1: The effect of schooling on wages
Some intuition:
Recall:
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Wages are assumed to equal expected output
Outputs are independent and identically distributed draws
⇒ wi1 is expectation of first period output given si and Xi
No part of ‘innovation’ in wages between first, second periods
(wi2 − wi1 ) can be forecast from information determining wi1
⇒ wi2 = wi1 + term depending on yi1 but orthogonal to si , Xi
⇒ estimated coefficients on si and Xi are the same in the first and
second and all subsequent periods
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Prediction #2: Unobserved characteristics
Recall: Bi in data but not observed by employers
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Note that other variables observable to employers
(si , Xi , and Zi ) could be correlated with Bi
Want to create a vector of variables orthogonal to employers’
information when worker enters labor market
Bi∗ : residual from a regression of Bi on all the other variables in the
data (si , Xi ) and the worker’s initial wage wi1
Bi∗ = Bi − E ∗ (Bi |si , Xi , wi1 )
Including wi1 conditions out employers’ information about Bi
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Caveat: measurement error in initial wage
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Prediction #2: Unobserved characteristics
Now add Bi∗ as a regressor to our wage equation:
wit
= αt + βt si + Xi γt + Bi∗ πt + εit
The question here is how πt will vary with experience
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Prediction #2: Unobserved characteristics
Specialize to case where B is a scalar
B ∗ is by construction orthogonal to the other regressors
⇒ πˆt =
cov (Bi∗ ,wit )
var (Bi∗ )
We can then write:
wit
= wit−1 + ζit
t
t
= wi1 +
ζiτ
τ =2
where ζit is the innovation in wages in each period
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Prediction #2: Unobserved characteristics
Bi∗ is orthogonal to wi1 by construction ⇒ π̂1 = 0 and:
cov (Bi∗ , wit ) = cov
Bi∗ , wi1 +
t
t
= cov (Bi∗ , wi1 ) + cov
t
t
=
τ =2
ζiτ
τ =2
Bi∗ ,
t
t
ζiτ
τ =2
cov (Bi∗ , ζiτ )
cov (Bi∗ , wit ) will “generally” be positive for every τ
⇒ πˆt will increase with t
⇒ if Bi∗ is correlated with ability, then the estimated effect of Bi∗ on
wages should increase with experience
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Prediction #2: Unobserved characteristics
Helpful to compare effect of characteristics market cannot observe (Bi∗ )
with effect of characteristics market can observe (si , Xi )
By construction, former play no role in wage determination, but their
estimated effect increases over time as the market learns about ability
by observing output
Latter play a declining role in the market’s inference process but have
a constant estimated effect
Key prediction of the model
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Prediction #3: Wage residuals
E (ζit |wit−1 ) = 0 ⇒ wages are a martingale: E (wit |wit−1 ) = wit−1
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You may be thinking: what is a martingale?
Not the focus of Altonji-Pierret test, so see paper for details
Fact that measured wage growth increases with experience implies
wages not a martingale; empirics focus on related prediction that
wage residuals (not wages) are a martingale
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Theory: Time-variant worker characteristics
Model thus far ruled out productivity growth with experience
Assume i th worker’s output in period t is Yit = yit + h(t)
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yit : part of output due to innate ability
h(t): part of output due to acquired skill
Assume output grows with experience by h(t) (OJT)
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h(t): deterministic, linear
Write down new wage equation:
wit
Williams (MIT 14.662)
= α0 + α1 t + β0 si + β1 si t + εit
Discrimination and learning
Spring 2015
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Empirical analysis
NLSY data
Panel data: wage dynamics for individuals
Focus on younger workers
Detailed experience measures
AFQT score as a measure of Bi
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Construct Bi∗
Many of the determinants of Bi are observable by the market, but we can
condition out other observables and the first period wage in order to
construct a measure Bi∗ that is not observed by employers:
ˆ i0
Bi∗ = Bi − Xi γˆ − δw
wit incorporates all information market has on worker’s ability
Caveat: wage may be measured with error ⇒ Bi∗ term will not be
completely purged of attributes observed by the market
Farber and Gibbons focus on AFQT, library card at age 14
(latter is a proxy for family background)
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Table 2
Table 2 tests Farber and Gibbons’ first and second predictions:
1
The estimated effect of schooling on the level of wages should be
independent of experience
2
Time-invariant worker characteristics correlated with ability but
unobserved by employers should be increasingly correlated with wages
as experience increases
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Table 2
Column (1): reports the means and standard deviations
Column (2): basic earnings regression
Column (3): adds AFQT and library card residuals
Consistent with the model:
1
No evidence that relationship between earnings and education varies
with experience
2
Interactions of the AFQT residuals with experience are positive
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Farber and Gibbons (1996): Table 2
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Prediction #3
Farber and Gibbons also analyze model’s third prediction:
Wage residuals should be a martingale
Not critical to understanding the Altonji-Pierret tests; see paper
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1
Preliminaries: Farber and Gibbons (1996)
2
Testing statistical discrimination: Altonji and Pierret (2001)
3
Affirmative action: Coate and Loury (1993)
4
Looking ahead
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Overview of Altonji and Pierret (2001)
Do employers statistically discriminate among young workers on the basis
of observable characteristics such as education and race, and as they learn
over time do they rely less on such variables?
Employer learning model as in Farber-Gibbons:
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Information common across firms
Labor market is competitive
Focus on variables such as race, which employers observe and could
be correlated with AFQT scores
Key idea: statistical discrimination with employer learning should
imply coefficient on AFQT will rise with experience whereas
(conditional on AFQT) coefficient on race will fall
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Differences: Altonji-Pierret and Farber-Gibbons
Very similar models; key differences:
1
Altonji-Pierret model specified in logs rather than levels
2
Whereas Farber-Gibbons orthogonalize Bi with respect to Xi and wi0 ,
Altonji-Pierret do not do this - they are interested in how changes in
relationship between Bi and wages over time affects coefficients on
Xi ’s such as race and schooling
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Altonji-Pierret model in one slide
Proposition 1: Assume schooling s is correlated with the initially
unobserved variable z (AFQT score). If we include z in the wage
regression with a time-varying coefficient, then as employers learn about
the productivity of workers the observable variable s (schooling) will get
less of the credit for an association with productivity as z can claim the
shifting credit.
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Do employers statistically discriminate on education?
Column (1): education, black, AFQT, educ-exp interaction
Column (2): adds AFQT-experience interaction
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Effect of AFQT rises from 0 (exp=0) to 0.0692 (exp=10)
Supports that employers learn about productivity over time
Coefficient on education-experience interaction declines:
supports that employers statistically discriminate on education
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Altonji and Pierret (2001): Table 1
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Do employers statistically discriminate on the basis of race?
A statistically discriminating firm might use race along with education
to predict the productivity of new workers
With experience, the productivity of the worker would become more
apparent, and compensation would be based on all of the information
available rather than just the information at the time of hire
Hence, if statistical discrimination based on race is important, then
adding interactions between t and z variables should make the
Black-experience interaction less negative
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If firms do not use (or partially use) race as information...
If race is negatively related to productivity, then the race gap should
widen with experience and adding AFQT-experience interaction will
reduce the race gap in experience slope
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Do employers statistically discriminate on the basis of race?
Race coefficients in Table 1 not consistent with statistical
discrimination
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Column (3): Black main effect becomes much less negative when
Black-experience interaction is added
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Suggests there is either not much difference in the productivity of black
and white men at the time of labor market entry, or that firms do not
statistically discriminate much
Race gap rises sharply with experience
Together, inconsistent with statistical discrimination based on race
Column (4): Adding AFQT-experience interaction decreases
Black-experience interaction
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Also inconsistent with statistical discrimination based on race
One interpretation: employers are obeying the law and not statistically
discriminating based on race
Paper also discusses alternative explanations
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Altonji and Pierret (2001): Table 1
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1
Preliminaries: Farber and Gibbons (1996)
2
Testing statistical discrimination: Altonji and Pierret (2001)
3
Affirmative action: Coate and Loury (1993)
4
Looking ahead
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Overview
Fryer-Loury (2005): recent overview of affirmative action
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“Regulations on the allocation of scarce positions in education,
employment, or business contracting so as to increase the
representation in those positions of people belonging to certain
population subgroups”
Holzer-Neumark (2000)
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Table 1: key executive orders, regulations, and court decisions
regarding affirmative action in the labor market
Reviews empirical studies of affirmative action policies
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(Selected) Empirics
Leonard (1984) on Executive Order no. 11246 in 1965
Chay (1998) on Equal Employment Opportunity Act of 1972
McCrary (2007) on a series of court-ordered racial hiring quotas in
municipal police departments
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Affirmative action has been controversial
One key question: can labor market gains from affirmative action policies
continue without these policies becoming a permanent fixture in the labor
market?
Coate and Loury (1993): how do affirmative action policies impact
employers’ stereotypes about capabilities of minority workers?
Break down negative stereotypes: could ⇒ permanent gains
Negative views about minority group are not eroded or are worsened:
policy would need to be maintained permanently
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Model set-up
Assume large number of identical employers
Assume large population of workers, randomly matched
Workers ∈ [B, W ], where share λ is W
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Employers
Assign workers to job 0 or job 1
All workers can perform satisfactorily job 0
Workers differ in qualification for job 1
Workers earn ω on task 1, 0 on task 0
Employers earn:
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xq > 0 for qualified worker on task 1
−xu < 0 for unqualified worker on task 1
0 for worker on task 0 (normalization)
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Employers
Employers don’t observe qualification prior to assignment
Observe worker identity ∈ [B, W ]
Observe noisy signal θ ∈ [0, 1] of worker’s qualification (test)
Distribution of θ depends on qualification, but not group
Fq (θ): probability signal does not exceed θ given qualified
Fu (θ): probability signal does not exceed θ given unqualified
fq (θ), fu (θ): density functions
ϕ=
fu (θ)
fq (θ) :
likelihood ratio at θ
Assume that ϕ (·) is non-increasing on θ ∈ [0, 1]
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Implies Fq (θ) ≤ Fu (θ) for all θ
Distribution of the signal for qualified workers first-order stochastically
dominates distribution for unqualified workers
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Employers
Employers’ assignment policies: thresholds for each group
Workers are qualified to perform task 1 only if they have made some
costly ex ante investment
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c: worker’s investment cost
G (c): fraction of workers with cost ≤ c
Cost distributions equal across groups
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Equilibrium
Equilibrium is a set of employer beliefs (about workers’ qualifications
in each group W and B) and workers’ investments that are
self-confirming
Discriminatory equilibrium is one in which employers believe that
workers from one group are less likely to be qualified
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Affirmative action
Consider an affirmative action policy that mandates that group
assignments to task 1 are made at equal rates
Ask whether introduction of such a constraint is sufficient to induce
employers - in the resulting equilibrium - to believe that workers’
productivities are uncorrelated with their group identity
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Coate and Loury: Conclusions
Coate and Loury conclude by saying their results give “credence to both
the hopes of advocates...and the concerns of critics.” There are
circumstances under which affirmative action will eliminate negative
stereotypes, but equally plausible circumstances under which it fail to do
so or even worse stereotypes.
More empirical work on these issues would be useful
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1
Preliminaries: Farber and Gibbons (1996)
2
Testing statistical discrimination: Altonji and Pierret (2001)
3
Affirmative action: Coate and Loury (1993)
4
Looking ahead
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Looking ahead
Rent-sharing
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14.662 Labor Economics II
Spring 2015
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